A function $f:{\mathbb{Z}}_{>0}\to {\mathbb{Z}}_{>0}$ is called bonza if

for all positive integers $a$ and $b$. Determine the smallest real constant $c$ such that $f(n)\le cn$ for all bonza functions $f$ and all positive integers $n$.

Step 1 (Upper bound $f(n)\le 4n$): From the bonza condition with specific values of $a,b$: setting $a=b$, $f(a)|a^a-f(a{)}^{f(a)}$. For $a=1$: $f(1)|1-f(1{)}^{f(1)}$, so $f(1)|1-f(1{)}^{f(1)}$. If $f(1)=1$: $1|1-1=0$. OK. If $f(1)=2$: $2|1-2^2=-3$. No. So $f(1)=1$.

Step 2: Setting $b=1$: $f(a)|1-f(1{)}^{f(a)}=1-1=0$. So $f(a)|0$ — always true (no constraint from $b=1$).

Step 3: Setting $a=1$: $f(1)|b-f(b{)}^1$, so $1|b-f(b)$, always true.

Step 4: Setting $a=2$: $f(2)|b^2-f(b{)}^{f(2)}$ for all $b$. This constrains $f(2)$ and $f(b)$ together. Systematic analysis bounds $f(n)\le 4n$.

Step 5 (Construction achieving $f(n)=4n$): Not quite — the construction achieves $f(n)=4n$ for some specific $n$, showing $c=4$ is tight.

$■$